Solve 3x+4+4y-xy^2=r^2

$$3 x + 4 + 4 y - x y ^ { 2 } = r ^ { 2 }$$

Solve for x (complex solution)

$$\left\{\begin{matrix}x=-\frac{4y-r^{2}+4}{3-y^{2}}\text{, }&y\neq -\sqrt{3}\text{ and }y\neq \sqrt{3}\\x\in \mathrm{C}\text{, }&\left(y=\sqrt{3}\text{ and }r=2\sqrt{\sqrt{3}+1}\right)\text{ or }\left(y=\sqrt{3}\text{ and }r=-2\sqrt{\sqrt{3}+1}\right)\text{ or }\left(y=-\sqrt{3}\text{ and }r=2i\sqrt{\sqrt{3}-1}\right)\text{ or }\left(y=-\sqrt{3}\text{ and }r=-2i\sqrt{\sqrt{3}-1}\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

Subtract 4 from both sides.

$$3x+4y-xy^{2}=r^{2}-4$$

Subtract 4y from both sides.

$$3x-xy^{2}=r^{2}-4-4y$$

Combine all terms containing x.

$$\left(3-y^{2}\right)x=r^{2}-4-4y$$

The equation is in standard form.

$$\left(3-y^{2}\right)x=-4y+r^{2}-4$$

Divide both sides by 3-y^{2}.

$$\frac{\left(3-y^{2}\right)x}{3-y^{2}}=\frac{-4y+r^{2}-4}{3-y^{2}}$$

Dividing by 3-y^{2} undoes the multiplication by 3-y^{2}.

$$x=\frac{-4y+r^{2}-4}{3-y^{2}}$$

Solve for x

$$\left\{\begin{matrix}x=-\frac{4y-r^{2}+4}{3-y^{2}}\text{, }&|y|\neq \sqrt{3}\\x\in \mathrm{R}\text{, }&y=\sqrt{3}\text{ and }|r|=2\sqrt{\sqrt{3}+1}\end{matrix}\right.$$

Steps for Solving Linear Equation

Subtract 4 from both sides.

$$3x+4y-xy^{2}=r^{2}-4$$

Subtract 4y from both sides.

$$3x-xy^{2}=r^{2}-4-4y$$

Combine all terms containing x.

$$\left(3-y^{2}\right)x=r^{2}-4-4y$$

The equation is in standard form.

$$\left(3-y^{2}\right)x=-4y+r^{2}-4$$

Divide both sides by 3-y^{2}.

$$\frac{\left(3-y^{2}\right)x}{3-y^{2}}=\frac{-4y+r^{2}-4}{3-y^{2}}$$

Dividing by 3-y^{2} undoes the multiplication by 3-y^{2}.

$$x=\frac{-4y+r^{2}-4}{3-y^{2}}$$

Solve for r (complex solution)

$$r=-\sqrt{4+4y+3x-xy^{2}}$$

$$r=\sqrt{4+4y+3x-xy^{2}}$$

Solve for r

$$r=\sqrt{4+4y+3x-xy^{2}}$$

$$r=-\sqrt{4+4y+3x-xy^{2}}\text{, }\left(y\geq \sqrt{3}\text{ and }x\leq -\frac{4\left(y+1\right)}{3-y^{2}}\right)\text{ or }\left(x\geq -\frac{4\left(y+1\right)}{3-y^{2}}\text{ and }y\leq \sqrt{3}\text{ and }y>-\sqrt{3}\right)\text{ or }y=\sqrt{3}\text{ or }\left(y<-\sqrt{3}\text{ and }x\leq -\frac{4\left(y+1\right)}{3-y^{2}}\right)\text{ or }\left(x=-\frac{4\left(y+1\right)}{3-y^{2}}\text{ and }|y|\neq \sqrt{3}\right)$$